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Chapter 10: Problem 71

Simplify. Assume that all variables represent positive numbers. $$\sqrt{8}-2 \sqrt{2}+\sqrt{12}$$

Short Answer

Expert verified
The simplified expression is \(\ 2 \sqrt{3} \).

Step by step solution

01

Simplify \( \sqrt{8} \)

Rewrite \(\sqrt{8} \) as \(\sqrt{4 \cdot 2} \). Then simplify it as \(\sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2} \). So, \(\sqrt{8} = 2 \sqrt{2} \).
02

Simplify \(\sqrt{12} \)

Rewrite \(\sqrt{12} \) as \(\sqrt{4 \cdot 3} \). Then simplify it as \(\sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3} \). So, \(\sqrt{12} = 2 \sqrt{3} \).
03

Combine like terms

Rewrite the original expression using the simplified radicals: \(\sqrt{8} - 2 \sqrt{2} + \sqrt{12} = 2 \sqrt{2} - 2 \sqrt{2} + 2 \sqrt{3} \). Then combine like terms: \(\ 2 \sqrt{2} - 2 \sqrt{2} = 0 \), so the expression reduces to \(\ 2 \sqrt{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). When simplifying square roots, it can be helpful to factor the number under the square root into its prime factors. This allows you to simplify more easily. \br\
Consider the expression \( \sqrt{8} \). To simplify, you rewrite it as \( \sqrt{4 \cdot 2} \). The square root of 4 simplifies to 2, so \( \sqrt{8} = 2 \sqrt{2} \)
Combining Like Terms
Combining like terms means to simplify an expression by summing constants and summing coefficients of the same variable. This is essential in solving algebraic expressions efficiently. \br\
In radical expressions, like terms have the same radical part. For example, \( \sqrt{2} \) and \( 2 \sqrt{2} \) are like terms because they both contain \( \sqrt{2} \). If you have \( 2 \sqrt{2} - 2 \sqrt{2} \), then these terms combine to zero, because their coefficients cancel each other out.
This step simplifies the expression and keeps it cleaner.
Radical Expressions
Radical expressions include roots (square roots, cube roots, etc.). Simplifying these expressions involves reducing the radicals if possible and combining like terms. \br\
For the original exercise: \( \sqrt{8} - 2 \sqrt{2} + \sqrt{12} \), the given radical expressions can be simplified as follows:
  • \( \sqrt{8} \) simplifies to \( 2 \sqrt{2} \)
  • \( \sqrt{12} \) simplifies to \( 2 \sqrt{3} \)
  • Next, combine like terms: \( 2 \sqrt{2} - 2 \sqrt{2} + 2 \sqrt{3} \)

    • Since the \( 2 \sqrt{2} - 2 \sqrt{2} \) cancels out, you're left with \( 2 \sqrt{3} \). \br The simplified radical expression is \( 2 \sqrt{3} \).

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